Improve Run Merging
• Reduce number of merge passes.
 Use higher order merge.
 Number of passes
= ceil(logk(number of initial runs))
where k is the merge order.
• More generally, a higher-order merge
reduces the cost of the optimal merge tree.
Improve Run Merging
• Overlap input, output, and internal merging.
DISK
MEMORY
DISK
Steady State Operation
Read from
disk
Write to
disk
Merge
Partitioning Of Memory
O0
O1
DISK
Loser
Tree
MEMORY
I0
I1
…
DISK
Ib
• Need exactly 2 output buffers.
• Need at least k+1 (k is merge order) input buffers.
• 2k input buffers suffice.
Number Of Input Buffers
• When 2 input buffers are dedicated to each
of the k runs being merged, 2k buffers are
not enough!
• Input buffers must be allocated to runs on
an as needed basis.
Buffer Allocation
• When ready to read a buffer load, determine
which run will exhaust first.
 Examine key of the last record read from each of
the k runs.
 Run with smallest last key read will exhaust first.
 Use an enforceable tie breaker.
• Next buffer load of input is to come from run
that will exhaust first, allocate an input buffer
to this run.
Buffer Layout
Output
buffers
F0
F1
F2
F3
F4
F5
F6
F7
F8
Input buffer
queues k=9
R0
R1
R2
R3
R4
R5
R6
Pool of free input buffers
R7
R8
Initialize To Merge k Runs
• Initialize k queues of input buffers, 1 queue per
run, 1 buffer per run.
• Input one buffer load from each of the k runs.
• Put k – 1 unused input buffers into pool of free
buffers.
• Set activeOutputBuffer = 0.
• Initiate input of next buffer load from first run to
exhaust. Use remaining unused input buffer for
this input.
The Method kWayMerge
• k-way merge from input queues to the active output
buffer.
• Merge stops when either the output buffer gets full or
when an end-of-run key is merged into the output
buffer.
• If merge hasn’t stopped and an input buffer gets
empty, advance to next buffer in queue and free empty
buffer.
Merge k Runs
repeat
kWayMerge;
wait for input/output to complete;
add new input buffer (if any) to queue for its run;
determine run that will exhaust first;
if (there is more input from this run)
initiate read of next block for this run;
initiate write of active output buffer;
activeOutputBuffer = 1 – activeOutputBuffer;
until end-of-run key merged;
What Can Go Wrong?
• k-way merge from input queues to the active output
buffer.
• Merge stops when either the output buffer gets full or
when an end-of-run key is merged into the output
buffer.
• If merge hasn’t stopped and an input buffer gets
empty, advance to next buffer in queue and free empty
buffer. There may be no next buffer in the queue.
What Can Go Wrong?
repeat
kWayMerge;
wait for input/output to complete;
add new input buffer (if any) to queue for its run;
determine run that will exhaust first;
There may be
if (there is more input from this run)
free input
initiate read of next block for this run; no
buffer to read
into.
initiate write of active output buffer;
activeOutputBuffer = 1 – activeOutputBuffer;
until end of run key merged;
• If merge hasn’t stopped and an input buffer gets
empty, advance to next buffer in queue and free empty
buffer. There may be no next buffer in the queue.
• If this type of failure were to happen, using two
different and valid analyses, we will end up
with inconsistent counts of the amount of data
available to kWayMerge.
• Data available to kWayMerge is data in
 Input buffer queues.
 Active output buffer.
 Excludes data in buffer being read or written.
No Next Buffer In Queue
repeat
kWayMerge;
wait for input/output to complete;
add new input buffer (if any) to queue for its run;
determine run that will exhaust first;
if (there is more input from this run)
initiate read of next block for this run;
initiate write of active output buffer;
activeOutputBuffer = 1 – activeOutputBuffer;
until end-of-run key merged;
• Exactly k buffer loads available to kWayMerge.
• If merge hasn’t stopped and an input buffer gets
empty, advance to next buffer in queue and free empty
buffer. There may be no next buffer in the queue.
• Alternative analysis of data available to kWayMerge
at time of failure.
 < 1 buffer load in active output buffer
 <= k – 1 buffer loads in remaining k – 1 queues
 Total data available to k-way merge is < k buffer
loads.
initiate read of next block for this run;
There may be
no free input
buffer to read
into.
• Suppose there is no free input buffer.
• One analysis will show there are exactly k + 1
buffer loads in memory (including newly read
input buffer) at time of failure.
• Another analysis will show there are > k + 1
buffer loads in memory at time of failure.
• Note that at time of failure there is no buffer
being read or written.
No Free Input Buffer
repeat
kWayMerge;
wait for input/output to complete;
add new input buffer (if any) to queue for its run;
determine run that will exhaust first;
if (there is more input from this run)
initiate read of next block for this run;
initiate write of active output buffer;
activeOutputBuffer = 1 – activeOutputBuffer;
until end-of-run key merged;
• Exactly k + 1 buffer loads in memory.
initiate read of next block for this run;
There may be
no free input
buffer to read
into.
• Alternative analysis of data in memory.
 1 buffer load in the active output buffer.
 1 input queue may have an empty first buffer.
 Remaining k – 1 input queues have a nonempty first
buffer.
 Remaining k input buffers must be in queues and full.
 Since k > 1, total data in memory is > k + 1 buffer
loads.
Minimize Wait Time For I/O To
Complete
Time to fill an output buffer
~ time to read a buffer load
~ time to write a buffer load
Initializing For Next k-way Merge
Change
if (there is more input from this run)
initiate read of next block for this run;
to
if (there is more input from this run)
initiate read of next block for this run;
else
initiate read of a block for the next k-way merge;